Journal of Experimental Botany, Vol. 53, No. 376, pp. 1919-1928,
September 1, 2002
© 2002 Oxford University Press
Source–sink partitioning. Do we need Münch?
Received 15 November 2001; Accepted 2 May 2002
INRA, Station Environnement et Grandes Cultures, F-78850 Thiverval-Grignon, France
Abbreviations: Source region parameters have subscript 0, sink 1 and sink 2 parameters have subscripts 1 and 2, respectively. Local concentrations of assimilate: S0; S1; S2. Michaelis constants for sinks 1 and 2: K1; K2; V1; V2. Resistance to concentration-driven flow within the common pathway of a 1-source/2-sink system: R0, and with the separate pathway associated with each sink: R1; R2. Activity fluxes either from source or to sinks: U0; U1; U2.
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Abstract |
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The simulation of phloem translocation by the Münch theory commonly uses resistances from sources to sinks: the resistances are therefore regarded as important in partitioning. Although resistance is generally a set constant, it is in fact strongly affected by viscosity, and thus the concentration of the transported solute. In this paper, the model of partitioning proposed by Minchin et al. was first corrected for variations in viscosity. The model was further modified, with the source considered as an activity of solute production rather than as a compartment concentration. When so defined, the source cannot differ from the sum of sink activities, largely outdating the source- or sink-limitation concepts. The corrected model confined the effect of resistances on the partitioning to low source activities. In the example of wheat grain filling analysed, such activities would be so low that they would correspond only to pathological conditions. In that case, the use of resistances in modelling is therefore just a mathematical burden, not even easily quantifiable since they are related to anatomical traits that are difficult to access. Leaving out resistances, it becomes easy to calculate the sink activities directly from the source activities, using an intuitive, accessible parametrization. The conditions for such a simplification are discussed.
Key words: Key words: Dry matter distribution, model, sink priority, sink size, sink strength.
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Introduction |
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Although attention has been given to partitioning for a long time, for example, wheat breeding for yield was improved by selecting for this character, the first crop simulation models were mostly directed to the study of nutrient capture, and led to the concept of non-limiting production. Only in a second step were so-called limiting factors (such as nutrient deficiencies, water stress or pest attack) considered in these early models (Moulin and Beckie, 1993). However, for both economic and environmental reasons, farmers are obliged to limit the inputs in crops, i.e. to grow their crops with limiting factors. High productivity is still required, with the addition that quality criteria for crops are now demanded. In consequence, greater attention has been paid recently to the partitioning of assimilates between sources and sinks, in order to explain how the plants manage when resources are limited. Older experiments indicated that the partitioning could vary with the overall level of nutrition, and may even lead to switches in sink priorities. The extensive shoot/root literature provides the best known examples of such a priority switch.
Among the first models for plant growth, one proposed by Thornley (1972) had already simulated the partitioning of assimilates using two physiological mechanisms: phloem translocation and assimilate use by the sinks. Phloem translocation is generally modelled according to Münch’s theory on the formation of a solution flow by an osmotically generated pressure gradient, so that the mass flow U1 to sink 1 may be obtained as following:
U1=S0x(S0–S1)/R1(1)
where S0 and S1 are the local concentrations of assimilate in source 0 and sink 1, respectively, and R1 is the hydraulic resistance from 0 to 1.
The transported assimilates are thereafter used by the sink, according to the Michaelis theory of enzyme kinetics,
U1=V1xS1/(K1+S1)(2)
where V1 is the maximum rate and K1 the affinity constant of the reaction.
Combining equations (1) and (2), the concentration S1 of the sink compartment, and then the flow U1, can thus be deduced from the concentration S0 of the source compartment and the three parameters R1, V1, and K1. Minchin et al. (1993) analysed the competition between two sinks, 1 and 2, respectively, for the assimilates exported by one source 0. They demonstrated that the flow rates, U1 and U2 to the sink compartments 1 and 2, respectively, can be calculated:
where R0 is the resistance associated with the common pathway from the source compartment to both sink compartments, and R1 and R2 are the resistances associated with the specific pathways to sink compartments 1 and 2, respectively. S0, S1 and S2 are the local concentrations of assimilate in source and sink compartments, while K1, V1, and K2, V2 are the Michaelis constants for sinks 1 and 2, respectively. The equations (3a) and (3b) must be solved using numerical techniques for a given set of parameters: R0, R1, R2, K1, V1, K2, and V2; with S0 as a variable.
Minchin et al. (1993) reproduced qualitatively some physiological results reported in the literature. Hence sink priority can be affected by girdling, as reported by Grusak and Lucas (1985), or a sink may be unable to profit from the suppression of its competitor, as indicated by Farrar and Minchin (1991). Unfortunately, the model of Minchin et al. (1993) often appeared limited to such qualitative applications, for two reasons. First, the variable S0 is as difficult to measure as it is to model. Moreover, the model of Minchin et al. (1993) could wrongly suggest that a source is adequately characterized as producing a stable concentration S0, whereas S0 is likely to vary as a result of the interaction between source and sink activities. Second, the hydraulic resistances are not easy to measure. Comparing the phloem to an impermeable tube, of radius x and length L, its resistance is often estimated by the Poiseuille law
where 
is the gas constant, T is the absolute temperature and 
is the viscosity of the transported solution. This means that the resistance, although specific to a source–sink pathway, is not a constant, because it will change in accordance with temperature as well as with viscosity, i.e. the solute concentration! Surprisingly, however, variation in viscosity is generally not taken into account in the papers dealing with resistance..
The present paper presents a revised version of the model of partitioning proposed by Minchin et al. (1993). Calculations are therefore made for a 1 source/2 sinks system (Fig. 1), but could easily be extended further. First, the original model is corrected for the effect of viscosity on resistance. It is then transformed with the source and sinks considered as activities, rather than compartments. The source is indeed much more intuitively characterized through its activity U0 of assimilate production than through its compartment concentration S0, which then becomes a result of the simulation, instead of input data. The resulting model is then tested for the sensitivity of its parameters, using either arbitrarily chosen parameters or the parameterization of partitioning during grain filling in wheat. In this example, it is suggested that the model may be further simplified by the useful omission of the resistances.
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Model correction
The way in which viscosity varies with temperature or concentration has not been fully determined. However, much data on this can be found in the reference literature. According to Weast (1972), the viscosity of water decreases as temperature increases. The effect on resistance is further amplified because temperature is also directly involved in the resistance calculation as reported in equation (4). For convenience, the variation of resistance with temperature is plotted in Fig. 2a using a relative scale: 100% represents 20R, the resistance at 20 °C. The variations remain moderate (about ±30%) in the 10–30 °C range, but greatly increase outside this range: at 0 °C, the resistance is twice that at 20 °C. Within a temperature
range of [0–42 °C], the resistance R fits the following cubic relationship:
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R=20R(–1.208x10–53+1.432x10–32–0.07318+2.07)(5a) This fit is by no means a physical explanation for the effect of temperature on hydraulic resistances, but it does correct for the effect with a resulting error of less than 1% within the physiological range of temperatures.
While phloem sap is a complex and variable mixture, sucrose is by far its main component. Data on pure sucrose solutions can thus be used to extrapolate the effect of sap concentration on viscosity and resistance (Fig. 2b). For sucrose concentrations that are physically possible, viscosity varied over several magnitudes. However, physiological concentrations are unlikely to increase beyond 1.5 M, still leading to a 10-fold variation in viscosity and, consequently, to variations in resistance. The variation of resistance with concentration is shown in Fig. 2b using a relative scale: 100% represents 3R, the resistance for a viscosity of 3 mPa s. This viscosity, used in Minchin et al. (1993), roughly corresponds to a 1 M sucrose concentration. As early as 1979, Magnuson et al. suggested taking account of the concentration effect on viscosity by using a polynomial fitting. In this paper, for a concentration S in the range of [0 M; 1.5 M], the resistance SR fits the following polynomial relationship:
SR=3R(0.685S4–1.0411S3+0.9513S2+0.1364S+0.3396)(5b)
As for equation (5a), this fit corrects the concentration effect with a resulting error of less than 1% within its physiological range. In the text that follows, the resistance is indicated as 3R, which is the resistance for a reference viscosity of 3 mPa s, but replaced in calculations by the corrected value SR.
The concentration varies along the phloem from S0 at the source compartment to S1 and S2 at the sink compartments 1 and 2, respectively (Fig. 1). A complete correction would therefore require the integration of this varying resistance into SR. However, such an integration of R over L from source to sink compartments, if at all possible, would be very difficult for two reasons: (i) S is obtained by resolving equations (3a) and (3b), which can only be done by numerical techniques for a given set of R0, R1, R2. In other words, there is no formal equation to be integrated for S. (ii) The Poiseuille law as reported in equation (4) should not suggest that the resistivity dR/dL is constant from source to sink compartments. Resistivity dramatically increases at the end of each sieve tube when passing through the pores, and it is unlikely that this effect could be described by a simple equation for the pathway from source to sink compartments.
Instead, the corrected value SR of a resistance will be calculated in this paper, using the mean concentration between the start and the end of this resistance. For R0, R1 and R2, the mean concentra tions used will be (S0+SF)/2; (SF+S1)/2 and (SF+S2)/2, respectively, where SF is the concentration at the end of the common phloem pathway from source to sink compartments (Fig. 1). According to equation (1),
U0=S0(S0–SF)/R0(6)
where U0 is the flux exported by the source, i.e. U1+U2. Because the concentrations S1 and S2, as well as the fluxes U1 and U2, must be numerically obtained, obtaining SF together with the resulting corrections for R0, R1 and R2 does not add any difficulty.
Parametrization for numeric simulations
Because equations (3a) and (3b) must be solved using numerical techniques, the parameters {R;K;V} of the sinks should be set before calculating the sink activities, varying the concentration S0 of the source compartment. These parameters are arbitrarily chosen for Figs 3 and 4, whereas wheat (Triticum aestivum) parameters are used as a quantitative example in Figs 5 and 6. The following section thus deals with the problems related to obtaining a fair parameterization, an important consideration in model development. The parameters were obtained from field assays using the variety Trémie in 1998 and 1999 at the INRA station, Grignon (France) during the fourth week after anthesis. The first sink ({K1;V1} parameters; see Fig. 1) is the ear at the grain-filling stage and the second sink ({K2;V2} parameters) is the polymerization of sucrose into fructans, forming temporary reserves in the stem internodes. Such reserves are eventually degraded to sustain grain-filling, and can thus be defined as a source activity. However, regardless of the net balance between storage and remobilization, fructan polymerization can be observed throughout the grain-filling period, either by measurements of enzyme activity (Bancal and Triboï, 1993) or by CO2 labelling (Gent, 1994). This activity is thus as a second sink, competing with grain-filling for sucrose. The source of carbohydrates (mostly net photosynthesis) will not be detailed in this paper. In the wheat example, the source activity U0 will be the flux out of the source compartment, regardless of any eventual internal regulation of this activity within the source compartment itself. Nevertheless, as a general rule, the model could work with source activities more strictly defined as the production of solutes to be used by the sinks. The resistance R0 (common pathway from the source to both sink compartments) and R1 (specific pathway to the sink compartment 1) were those of the leaves and the stem, respectively. The resistance R2 is associated with the pathway from the sieve tubes to the fructan storage sites, both of them situated in the stem.
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During the fourth week after anthesis, grain-filling was in the so-called linear phase, which meant that no change in its rate could be attributed to plant ageing. If one half of the plants were removed from the field, the net photosynthesis rate of the remaining plants largely increased, but the rate of grain-filling only slightly increased, suggesting that this process was not far from saturation. This recorded grain-filling rate was corrected for 25% growth respiration to obtain an estimate of V1 (125 mg of dry mass d–1, or 4.3 nmol of sucrose s–1 per culm). According to Jenner et al. (1991), the grain-filling rate, mainly due to starch deposition, is regulated through sucrose import by the ear. Numerous studies using in vitro cultures of wheat ears have been published (Jenner, 1970; Jenner and Rathjen, 1978; Gifford and Brenner, 1981; Armstrong et al., 1987). The in vitro growth rates published for grain were fitted to the corresponding sucrose concentrations in the culture medium through Michaelis equations with affinity ranging from 30 mM to 100 mM. The average value, 65 mM, will be used for K1. This affinity does not reflect that of any ‘key’ enzyme, for example, starch polymerase or transporter, but rather that of a whole organ over its exogenous nutrition, which is the topic of this paper.
Because 6kestose is by far the most abundant fructan in the stem, its synthesis by the sucrose fructan fructosyltransferase can be considered as representative of the whole reserve formation under field conditions (Duchateau et al., 1995). Unfortunately, the in vitro characterization of this enzyme does not indicate any trend to saturation even at very high sucrose concentrations (Simmen et al., 1993). It may still fit the Michaelis equations providing that a very high value for K2, such as 3000 mM, is used so that S2 always remains much lower than K2. The sucrose consumption rate for 6kestose formation by cell-free extracts of plant stems was then used to estimate V2. This rate was measured at 0.76 nmol s–1 per culm, using a 500 mM sucrose concentration; if K2=3000 mM, V2 is thus 5.3 nmol s–1 per culm.
The parametrization of resistance is by far the most speculative component of the model, but it is probably unavoidable if the aim is to find a single parameter to represent the complete pathway between organs. According to Fisher and Gifford (1987), the bundle cross-sectional area is constant over the length of organs, this parameter was therefore measured at a single point to estimate the corresponding organ resistance. However, because there was not a single leaf but several leaves, the estimate of R0 resistance of the leaves needed prior consideration. The flag leaf contained about 40% of total leaf nitrogen, so it was assumed that it provided 40% of photosynthate (Sinclair and Horie, 1989). It was further hypothesized that the conductance of phloem bundles was proportional to the flow rate they transported, meaning that the inverse of flag leaf resistance was 40% of 1/R0. To estimate resistances, three plants were sampled from the field, and the stems and flag leaves were hand-cut into thin slices. The slices were then digested in hypochlorite and stained. Each slice exhibited about 50–60 vascular bundles, which were individually micro-photographed, and the pictures digitized. The cross-sectional area for every phloem cell was automatically recorded using image analysis procedures. However, the phloem bundles varied considerably in size and location. In stems, as reported in Fisher and Gifford (1987), large vascular bundles were interior whereas small bundles were adjacent to chlorenchyma tissue. A wide range of bundle sizes was observed in leaves, with some of the smallest directed transversally rather than longitudinally. If this variation in bundle size is related to differences between the organs they irrigate, then not all of them should be involved in R0 and R1 calculation. When only the biggest bundles are considered, the resistances increase. Only these last values were used in this paper, and the Poiseuille law was then applied to each selected phloem cell according to equation (4) and using L=0.2 m for the leaves and L=0.4 m for the stems. The temperature was 20 °C and the viscosity 3 mPa s, as indicated in Minchin et al. (1993), leading to R0=3.9 Tmol s m–6 and R1=7.5 Tmol s m–6, respectively. However, the estimates for R0 and R1 were obtained from the cross-sectional area of phloem cells when the hydraulic resistance is greatly enhanced at the end of sieve tubes. It is clearly not possible to measure the number, diameter separation and direction of all the sieve pores in a plant. Instead, the preceding resistances were increased by 50% to take into account the greater resistance at pores at the end of sieve tubes (Sheehy et al., 1995), so that R0 and R1 are 5.7 and 10.9 Tmol s m–6, respectively. As indicated earlier, the resistance R2 is that from one part of the stem to another part of the stem. Of course, specific phloem strands can run towards particular tissues. Nevertheless, R2 is set at zero in this paper. This statement suggests that the sinks are serially connected to the source, which is clearly a simplification. Thus the simulation demonstrates that it has paradoxically little effect on the partition between sinks.
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Correction for variable resistance
In this section, an investigation was made of the partitioning of assimilates exported from one source to two unequal sinks. In Fig. 3 the sink parameters {R;K;V} are arbitrarily chosen {10;100;50} and {200;100;50} so that the sinks only differ by a very large difference in resistance. The resistance of the common pathway R0 is set at zero (otherwise in this example R0 will just dampen the observed effects). In Fig. 3A, U1, the activity of the sink 1, is plotted against the concentration S0 at the source compartment, using three different calculation hypotheses. If resistance is not taken into account, then the concentrations will not change from source to sink compartments, and the curve (dashed line) will be hyperbolic, like any Michaelis kinetic. If constant resistances are involved, their effect is to decrease the concentration in the sink compartment relative to that in the source compartment: S1<S0. However, for high S0, the resulting S1 is high enough to saturate the sink activity anyway. Consequently, the effect of the resistances on sink activity is important when concentrations are low, but it is essentially overcome when they increase: the resulting curve (dotted line) shifts from an hyperbola to a sigmoid. Now if the resistances are corrected for viscosity as indicated in this paper, the resulting curve (solid line) is situated between the other two. The correction indeed leads to a decrease in the resistances at low concentrations, where they affect the sink activities. In this example the way resistances are taken into account only has an effect for low S0 (see insert in Fig. 3A): beyond 0.5 M the curves overlap. However, as previously observed, concentration is an inadequate way of characterizing a source, which is better achieved using source activity.
Characterizing the source by activity rather than compartment concentration
Using the corrected model, it is not possible to calculate S0 from U0; however, numeric resolutions for U1 and U2 can be obtained, starting from S0. The relationship between S0 and U0 is then easily deduced according to the mass conservation law, since the flux exported by the source U0 cannot differ from the flux imported by the sinks U1 + U2, so
U0=U1+U2(7)
Consequently, U0 is obtained numerically from S0, and it is not difficult to deduce S0 from U0 using a computer program, but it should be noted that although the concentration in the source compartment can be increased virtually to infinity, this is not possible for source activity. From equations (2) and (7), U0<V1+V2. When U0=V1+V2, the concentrations are infinite, so a source activity U0 higher than V1+V2 falls outside the model. This will be discussed later. Note that in the example shown in Fig. 3, V1+V2=100 nmol s–1, which is thus the maximum possible U0 source activity. The relationship between source activity U0 and concentration S0 of the source compartment (Fig. 3B) is clearly not linear, but rather hyperbolic with an asymptote for U0=V1+V2. For low and medium activity, up to 75 nmol s–1, the way resistances are taken into account for the calculation of sink activities strongly affects this relationship. S0 increases more rapidly with U0 using constant resistances (dotted line) than using variable resistance (solid line), the smaller increase being obtained when resistance is not taken into account (dashed line). However, the resulting concentrations for these low and medium activities are low: less than 0.5 M. For activity greater than 75 nmol s–1, the importance of the calculation hypothesis on the resulting concentrations declines progressively, and the lines overlap. The 1.5 M concentration, which can be regarded as a physiological maximum, is obtained for 93 nmol s–1
Consequently, despite the fact that the way resistances are taken into account may appear quite unimportant when U1 is related to S0 (Fig. 3A), it clearly is meaningful in the relationship between U0 and U1 (Fig. 3C). If the resistances are not taken in to account (dashed line), then both sinks are equivalent and, consequently, U1 is just half of U0. On the other hand, constant resistances clearly advantage sink 1, whose activity U1 is more than half of U0 (solid line). This advantage increases up to approximately U0=40 nmol s–1, it rapidly declines thereafter and becomes negligible only when beyond approximately U0=80 nmol s–1. The effect of variable resistances (dotted line), while of less importance, is essentially the same as that of constant resistances. It is thus visible over almost the whole source activity range. In the preceding example, the importance of the resistances for the calculation of sink activities appears more clearly using U0 rather than S0 to characterize the source, but the converse could also be observed using other {R;K;V} parameters (data not shown).
Sensitivity of the corrected model to its parameters {R;K;V}
For Fig. 4, as for Fig. 3, the resistance R0 of the common pathway is arbitrarily set at zero, and V1+V2=100 nmol s–1. The sinks differ by one parameter only, i.e. either R ({10;100;50} for sink 1 and {200;100;50} for sink 2), or K ({10;30;50} sink 1 and {10;600;50} sink 2) or V ({10;100;95} sink 1 and {10;100;5} sink 2). The partitioning coefficient for sink 1 (U1 /U0 ratio) is plotted against the source activity U0. The lines obtained for S0 concentrations in the 0.1–1.5 M range are drawn in bold; note that the corresponding U0 source activity range is not the same for the three curves. This range is thought to correspond to physiologically relevant situations (see Discussion). The left part of the figure, drawn in thin lines, corresponds to very low S0 source concentrations, covering a very large range of U0 source activity in the example shown, due to the choice of parameters. This is not always true, and will be discussed later. The parameters were chosen largely to advantage sink 1, therefore its partitioning coefficient is always higher than the 50% (which would be partitioned if the sinks were equivalent).
The three parameters do not determine the partition in the same activity range. The effect of resistance, while maximum at medium source activity for absolute sink activity (Fig. 3C; dotted line), is maximum at low source activities for the partitioning coefficient (Fig. 4; solid line). The resistance effect progressively declines with greater source activity. By contrast, the K parameter, which mainly determines the partition at medium source activities (Fig. 4; dashed line), has no influence on partitioning, either at low or at high source activity. Once more the resistance effect dominates at low source activity, in this case leading to a 50% partitioning coefficient because the resistances are equal. At medium source activity, the K parameter affects the partition, here favouring sink 1 until concentrations exceed K2. Higher activities lead to a saturation trend for both sinks, in this case at the same V activity, and the partitioning coefficients therefore return to 50%. Lastly, a difference in the V parameter (Fig. 4; dotted line) affects the partitioning at high, but not at low activities. This is not surprising: when concentrations are high, the sinks saturate anyway, and their activities approach their V parameter, which logically determines competition. On the other hand, V has very little influence at low concentrations and, consequently, at low source activities, leading to a 50% partition because the R and K parameters are equal here.
Model adaptation for a specific purpose
In the preceding examples, some particular properties of the model were emphasized using arbitrarily chosen parameters. In the following section, the model is worked using parameters from partitioning in wheat. This section does not set out to provide a complete description of wheat carbon metabolism, but just uses wheat to illustrate how the model of carbon partitioning may be used: because the parameter effects vary over the range of source activity, the model can usefully be modified for such specific purposes. Figure 5 indicates the result of simulating either grain-filling rate U1 (Fig. 5A) or partition to grain-filling U1/U0 (Fig. 5B) when the U0 source activity varies. Lines drawn in bold correspond to the S0 concentrations in the range 0.1–1.5 M.
The solid lines indicate the simulation produced by the complete {R;K;V} model. Physiologically relevant S0 were obtained in the 2.0–5.5 nmol s–1 U0 activity range, i.e. 20–60% of V1+V2. The grain-filling rate U1 increases linearly with U0 until this latter reaches approximately 3 nmol s–1, then U1 progressively saturates. The partition to grain-filling is very small at very low source activities, corresponding to S0 concentrations less than 0.1 M. But the partition then increases rapidly, up to 90% with U0 from 1 to 3 nmol s–1, progressively declining thereafter to 70% at 5.5 nmol s–1 U0 activity. This decline continues thereafter, yet is meaningless since the corresponding S0 concentration is physiologically too high.
Such results can be compared with data from plants. For instance, U0 activity in control plants ranged within 3.5–5.0 nmol s–1 which, according to the model, is approximately 0.25–0.80 M for the S0 concentration. Such comparisons, however, are not the topic of this paper. Rather, the complete {R;K;V} model will now be compared with other simple models designed for this specific purpose. For instance, several classic partition models calculate, according to the overall plant status, a potential activity for each sink, for instance U'1 and U'2 for sinks 1 and 2, respectively. The models then compare the sum of potential sink activities to the source activity to obtain the actual sink activity. For sink 1, it will be
U1=U'1U0/(U'1+U'2), or U1=
1U0
where 1=U'1/(U'1+U'2), independent of U0 activity, is actually a partitioning coefficient to sink 1. Sinks eventually become completely saturated, so that U1 activity calculates as
U1=min [1U0; U'1]
Simulations by such a partition model, using 1=90% and U'1=V1 defined from the {R;K;V} model, are drawn in Fig. 5 (dotted lines). The resulting curves are similar to that of the {R;K;V} model in the 1–3.5 nmol s–1 range of U0 activity. For greater activities, however, the progressive saturation of sink 1 is not taken into account. Consequently, both sink 1 activity and partition are overestimated. Unfortunately, this occurs within the natural range of variation for U0 activity. Some partitioning models also involve transport and resistance from source to sinks. Such models would produce curves that are closer to the {R;K;V} model than that shown, especially in the left part of the Fig. 5B. In the case of wheat however, this part of the curve is of little interest.
In wheat, the resistance mostly affects partition in the range of U0 activity where the S0 concentration is lower than 0.1 M, considered as physiologically too low, so a model using Michaelis-type sinks connected without resistance was also tested (dashed lines). The {K;V} parameters are the same as those of the complete {R;K;V} model (solid lines). However, as previously indicated, the concentration does not change from source to sinks S0=S1=S2=S. Consequently, from equation (7):
U0=[V1S/(K1+S)]+[V2S/(K2+S)](8)
So S (and thus U1) can easily be calculated from U0. Even though this last model does not describe the transport from source to sinks, it is much more readily usable than the former model. In Fig. 5B, the simulated partitions are very close except at low, unrealistic U0 source activities. The simulated activities of sink 1 (Fig. 5A) differ so little that this could not have been detected using experimental, noisy, data.
Because it is difficult to assign a precise value to the resistances (see Materials and methods), Fig. 6 further explores the resistance effect in the specific function of grain-filling in wheat for three S0 concentrations, 0.1, 0.5 and 1.5 M. In Fig. 6A, the resistance to sink 2, R2, is varied from 0 to 100 Tmol s m–6 in order to study the partitioning coefficient to sink 1 in both the complete {R;K;V} model and its simplified version in which no account is taken of resistance. In the latter (dashed lines) of course, the R2 value does not alter the partition, which is only affected by the {K;V} parameters and the S0 concentration. The partition coefficient to sink 1 is thus constant at 94%, 83% and 70% for S0=0.1, 0.5 and 1.5 M, respectively. This coefficient is very similar to that of the complete {R;K;V} model (solid lines) when R2=0 Tmol s m–6 reaching 91%, 84% and 71% for S0=0.1; 0.5 and 1.5 M, respectively. It hardly increases with increasing R2, to 94%, 86% and 73% for S0=0.1, 0.5 and 1.5 M respectively for R2=100 Tmol s m–6, i.e. nine times the R1 value! In Fig. 6B, the resistance R2 is set to zero and the resistance R1 is varied. The partition coefficient to sink 1 obtained using the complete {R;K;V} model (solid lines) is slightly higher than that obtained using the simplified version (dashed lines) for R1=0 Tmol s m–6. This coefficient then decreases with increasing R1, but very slightly for either medium or high S0 concentration, so that the results of both models remain close, at least up to R1=100 Tmol s m–6. For the lower S0 concentration, however, the models differ more rapidly. With the actual value measured for R1 (10.9 Tmol s m–6), the partitioning to sink 1 is already 3% underestimated and the error rapidly increased with increasing R1. An error less than 10% (obtained for any R1<26.9 Tmol s m–6) is of course negligible in the estimation of sink 1 nutrition, but it could be important when focusing on sink 2 nutrition. However, apart from the case of continuous very low source activity, the simplified version provides an easy and useful way to model the partition from source activity. Even in such a case, the results of the complete {R;K;V} model do not require high precision in the resistance measurement.
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Discussion |
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As far back as 1967, Warren-Wilson tried to define the sink strength as the product of a size and a specific activity, i.e. by two distinct parameters, which would be local properties of a plant organ (Warren-Wilson, 1967). However, over the years, the sink strength became defined by a single parameter that various authors widely discussed. Thus a debate entitled ‘Sink strength, what is it and how do we measure it?’ was edited by Farrar (Farrar, 1993). It is clear, however, that it is very difficult, by using a single parameter, to describe the switches in partition that can frequently be noted when source activity varies.
The Münch theory applied to model phloem translocation (Christy and Ferrier, 1973) introduced a new variable into the source/sink world. According to this theory, sources and sinks are defined by compartment concentrations. Logically, Magnuson et al. (1979) therefore describe the activities of sink as enzyme kinetics, because such activities are easily obtained from a concentration using the Michaelis equation. A considerable advance in the understanding of source/sink relationships was thus obtained. Minchin et al. (1993) indeed combined the Münch and Michaelis equations to obtain a model in which the partition is explained by plant architecture and local properties of the sink compartments. Models published before their paper referred, to a greater or lesser extent, to the functional balance hypothesis of Thornley (1972) to explain the shoot/root ratios. According to such models, what is in one respect a sink (e.g. for carbon metabolism) is a source in other respects (e.g. for nitrogen metabolism). Close links between these metabolic pathways explained the partition. Such models were therefore unable to explain the partition to a sink that provides nothing to the rest of the plant, such as the two examples used in this paper, grain-filling and temporary reserves.
In addition, the Michaelis parameters {K;V} also appear to be useful for the proper characterization of the sinks. The V parameter is both a sink size (linked to quantitative properties of an organ, as emphasized by Warren-Wilson, 1967), and a sink potential (a mathematical limit for an activity which would be obtained under infinite nutrient provision, as suggested by Wareing and Patrick, 1976). Being a sink potential, V could be measured by fitting the actual sink activity to a Michaelis model using a range of source activities. Being a sink size, V can be obtained by every existing growth model. The best way of obtaining it is likely to be through the quantitative properties built by the previous growth of the corresponding organ (Munier-Jolain and Ney, 1998). So the partition will be included in growth models. The K parameter on the other hand explains the sink hierarchy observed in response to source variation (Wardlaw, 1990). Being linked to the overall metabolism of the sink organ, K will change along with ontological evolution in metabolism. On the other hand, since it roughly reflects enzyme affinities, K will be little affected by the actual physiological conditions, such as temperature, water status or nitrogen nutrition. Thus, this parameter could be obtained either from the bibliography or from specifically designed experiments, unrelated to even the model application.
These useful attributes of the {K;V} parameters have meant that the Minchin et al. (1993) paper has been referred to by several other authors, such as Sheehy et al. (1996) or Lemaire and Millard (1999). However, they are theoretical descriptions rather than usable models. The characterization of the source as a compartment may contribute to this misuse. Calculations can, in fact, only be made by using the concentration of the source compartment as a variable and not by using the source activity. However, characterizing the source as an activity is much more convenient than characterizing it as a compartment concentration. Firstly, while activities are easily obtained, concentrations are difficult both to measure accurately in vivo as well as to model independently. Secondly, the concentration in the source compartment is itself a result of source/sink interactions, even in those cases where the activity of a source is regulated through the concentration in their compartment. It is proposed here to link the source and sink activities according to the mass conservation law: the flux formed by the sources is the flux imported by the sinks. In other models, these two fluxes can differ by variation in the content of so-called labile compounds. However, temporary storage, such as that of starch in the leaves, is a sink in our definition, thus the labile compounds are only those located within the phloem bundles. According to Schnyder (1993) such ‘en-route’ compounds would sustain sink activities for less than 1 h; they can thus be left out of the partition model at the day or week scale.
Now equalizing source and sink activities leads to some quite surprising observations. It is no longer possible to tell whether a source is in excess relative to the sinks (or vice versa) because the fluxes are equal. Neither source nor sink limitation, nor limiting or non-limiting conditions can be emphasized. According to the Michaelis equation, any sink activity is always limited both through sink parameters and through source concentration, i.e. there is always co-limitation between source and sink (Kacser and Burns, 1973). Another paradoxical observation is that the source activity becomes limited by the V-sum, and thus by sink parameters! In fact, the V-parameters reflect the sink sizes, which are built up by sink growth (Black, 1993), that is, through source activity during the previous weeks. Relationships between sources and sinks thus appear not only through nutrition, but also through ontogeny. However, this balance can be broken: if a sink-bearing organ suddenly disappears, do the source activities become excessive? Many papers report such pruning experiments, and they always show that the plant adapts rapidly to the new situation. Models already exist which can take such accidents into account. Thornley (1995) for instance, inhibited the source when solute concentrations increase; Sheehy et al. (1996) instead turned their attention to excretion sinks with low affinity. Both these mathematical solutions, agreeing with physiological observations, avoid any accidental excessive accumulation of assimilates, but are essentially turned off in normal cases.
Although easily compatible with computer procedures, the calculation of sink activities is more complicated starting from source activity rather than source concentration, which comes mathematically from the use of resistance. Resistance is needed to explain the transport from source to sinks physically. However, in the Materials and methods, the difficulties in obtaining a fair value were indicated: Which bundles are involved in the pathway from a source to a sink? Is the resistance constant along a bundle (clearly not if variations in viscosity are taken into account)? How precisely is the effect of pores quantified? Moreover, the assumptions made in order to apply the Poiseuille law are far from realistic: the phloem cells are not impermeable cylindrical tubes. Clearly only a crude estimate of resistance can be obtained in such a manner. These difficulties led Maillard et al. (2000) to regard the description of resistance as too difficult to be used at crop level. On the other hand, the Münch theory is itself criticized as being too simple (Tyree and Dainty, 1975; Milburn and Kallarackal, 1989; Kargol et al., 2001). Alternative models need still more parameters, which are no easier to obtain. A correction of resistance for the variations in viscosity is now proposed. Many of the previous reports, using a single constant viscosity (generally that of pure water!), should at least be re-examined. If they somehow agree with data, it may be because the transport simulation is not always a critical part of the model.
In fact, resistances only affect the partition at low source activities; correcting them for viscosity variation further reduces the range over which they have an effect. However, source activities that are too low should not be taken into consideration because they would correspond to unacceptably low concentrations. As far as is known, phloem concentrations lower than 0.1 M have never been reported; and it is thought that they could only be considered for non-photosynthesizing, reserve-less, i.e. dying plants. In the case of wheat-grain filling, it is demonstrated that, since the resistance effect is limited to this non-physiological range of source activity, it can be left out of the simulation. This conclusion is in good agreement with experiments suggesting that transport capacity is not a limiting factor. Wardlaw and Moncur (1976) showed that cutting one half of the stem phloem does not affect grain filling. According to Wang et al. (1993) an excess transport capacity may actually be built, under the control of the sink itself.
Without resistance, the calculations become much simpler and intuitive. But while resistance is not a required factor in the case of wheat grain-filling, this is not always the case, as indicated in Figs 3–4. So two tests should be done using the complete {R;K;V} model before simplifying it. First, the strength of the resistance effect should be tested throughout the range of source activity. This, of course, depends on the R parameters themselves: resistances greater than 10 Tmol s m–6 will likely modify the partitioning for low, but still possible, source activities. The K parameters are also involved: the resistance effect is more important when sinks are far from being saturated. Hence sinks half-saturate only when their concentration reaches K. The lower the K value, the wider the source activity range in which sink activities can be decreased by resistance. The second test is to verify the left part of the curves, which sometimes includes a very large range of source activity, but corresponds to phloem concentrations, that are too low to have any physiological meaning. The extent of this excluded range will be sensitive to the same parameters R and K: the lower the K value, the wider the range of source activities in which concentration will be lower than K, and possibly less than 0.1 M.
Resistances are likely to remain key factors where numerous similar sinks interact with numerous similar sources such as fruits on a tree, which can explain the preferential pathways used by assimilates. Other examples of resistance involvement can be found where source activity is low and/or resistances are increased, such as bud nutrition in early spring or pest attacks. Even when resistances cannot be avoided, however, the partitioning will respect the law of mass conservation, and the consequent relationships between source activity and sink parameters will still apply.
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This work was carried out with the technical assistance of J Jean-Jacques and of the team of the Unité Expérimentale de Grignon, INRA, F-78850 Thiverval-Grignon, France. Preparation of microscope slides for the measurements of vascular bundles was done under the supervision of D Chriqui, Laboratoire CEMV, Université Pierre et Marie Curie, F-75252 Paris cedex 05, France.
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